Advice on Success in University Mathematics Courses

Many students ask me how to study. Particularly
in freshman mathematics courses, students may experience a sort of
culture shock: university courses have far more depth and a faster
pace than high school courses. Here are my recommendations
to students.

Solve lots and lots of math problems. If you solve
problems, you will learn mathematics. If you don’t solve
problems, you won’t learn mathematics. This is far and
away the most important determinant of success in math courses, and it
is universal. Solving problems is the way all human beings learn
mathematics, be they six-year-olds or professional mathematicians.
Your professor has probably assigned a number of problems, carefully
selected as instructive. Solve all of these problems—plus
additional, unassigned problems as needed, possibly from supplemental
sources—unaided insofar as possible. Resist the temptation
to look up a solution to a problem on the Internet! By obtaining
someone else’s solution instead of thinking up your own, you
deprive yourself of a crucial element of your mathematical development,
as well as development of habits such as perseverance. If, after
protracted effort, you’re completely stuck, ask your professor
about the problem. He or she may be able to point you in the
right direction with an appropriate hint while still allowing you to
develop your own ideas.

Master the prerequisite material. Later mathematics
builds upon earlier mathematics: the further you go in the subject,
the more abstract and subtle it becomes. If you have a solid
foundation in the background material, your experience will be of new
worlds opening up before you. If, however, your understanding
of the prior mathematics is particularly weak, learning subsequent
mathematics is likely to be an insuperable task. This problem
notably arises in freshman calculus classes, where the unprepared student
has to struggle to understand every little algebraic manipulation
in the course of a solution to a problem. It’s as if
someone didn’t understand a word of Russian but decided to
read untranslated Tolstoy by looking up every word in a bilingual
dictionary. Not too likely to yield a real understanding
of—or appreciation for—a literary masterpiece. If
you do take a math class without having assimilated the antecedent
mathematics, allocate ample time for review and expect to have to work
twice as hard.

Schedule sufficient study time. A one-semester math
course is likely to require 150 to 200 hours of studying and solving
problems outside of class. This work must be evenly distributed
throughout the semester.1 It
is important to avoid distractions while studying mathematics (and in
the classroom): engaging in a secondary task impedes the acquisition
of so-called “flexible knowledge,” the type needed to
solve novel problems.2
Beware the temptation to curtail your efforts when you think
you’ve fully understood the material. There is a
natural human proclivity to overestimate our own proficiency.3 Math students with a superficial
understanding of the material often lament, “I understood
the concepts, but I got lost when I tried to solve problems
on the exam.” This statement betrays a fundamental
misconception. Understanding the concepts is not the objective
of a mathematics course. I understand the concept of running a
four-minute mile. I understand the concept of speaking fluent
Welsh. I understand the concept of writing a novel that wins
the Man Booker Prize. Sad to say, I can’t actually do any
of those things. Understanding the concepts is, as a matter of
fact, a pretty trivial accomplishment. The true test of your
understanding is the ability to solve problems you’ve never seen
before using the mathematical ideas and techniques of the course.

Attend all class meetings without fail.
You will benefit from preparing for class in advance, for example, by
reading ahead in the textbook. Incidentally, it’s impolite
to arrive late.

Take careful notes during lectures; then review, annotate,
and assimilate your lecture notes. Some students say
it’s difficult to concentrate on a lecture while simultaneously
taking notes. The reason it’s difficult is also the reason
you should do it. Passively listening is easier than actively
engaging, but you learn less. Taking notes reinforces understanding
while the lecture is in progress, by forcing you to formulate what you
are seeing and hearing in your own words (in addition, obviously, to
providing you with a resource for later study). Merely recording
the lecture or taking photographs of the blackboard robs you of this
benefit; even taking notes on a laptop is less efficacious than taking
notes with pen and paper.4

Read the textbook slowly and carefully. Ask
questions about parts you don’t understand. Auxiliary
reading from books in the library is also beneficial. Remember
that reading a mathematics textbook is not like reading the newspaper:
you may need to pore over each sentence, wherein each word is apt to
be significant. Treat the occasion as a conversation between
you and the author. Jot down notes, fill in skipped steps, draw
accompanying illustrations. Cover up the proof of a theorem or
the conclusion of an example and see if you can produce the argument
yourself. You should read and digest the textbook material before
attempting to solve the relevant problems.

Form study groups with other students. It will make
studying more pleasant if it becomes a social activity. By working
with others, you create incentives for yourself to prepare. It
will help you understand difficult topics to get an explanation from a
classmate who has mastered the point in question. It will solidify
your understanding to explain topics to your peers. Communicating
mathematics is a skill worth cultivating. If you can’t
explain some idea to a fellow student who doesn’t understand it,
that might suggest that you don’t truly understand it yourself.

Attend office hours. There is great value to
individual attention from your professor, who is apt to have long
experience resolving common mathematical difficulties, but may not
be able to divine the nature of your misunderstandings unless you ask
about them. It helps to come prepared, but you shouldn’t
let concerns about that stop you. And don’t feel you are
somehow intruding. Office hours are your time.

Avail yourself of free tutoring. Most
universities—certainly my own—provide tutoring by graduate
students or advanced undergraduates; you should take advantage of this
valuable resource. You can get times and locations from the math
department office. An important caveat bears emphasis.
It is educationally harmful, not beneficial, to get a tutor simply to
show you an answer to a math problem you couldn’t solve without
teaching you the underlying mathematical principles. If you cannot
solve a problem—and certain problems can require a great deal of
thought and effort—that is an indication that you have not truly
understood and absorbed the relevant mathematical concepts as they
arise in practice. To simply “get help with a homework
problem” without achieving an understanding of the mathematical
principles relevant to it and other, similar problems is to treat the
symptom and not the disease.

Solve lots and lots of math problems. My first
recommendation is also my last recommendation; this vital point
bears repeating. You won’t become proficient at serving a
tennis ball merely by watching Serena Williams; you won’t become
proficient at playing the piano merely by listening to Glenn Gould;
you won’t become proficient at mathematics merely by watching
a mathematician solve problems. You have to do it yourself,
diligently and repeatedly. The reward is inestimable: not
simply a good performance on an exam, but a more powerful brain.5

If the way which I have pointed out as leading hither seems
exceedingly hard, it can nevertheless be discovered. Needs
must it be hard, since it is so seldom found. How would it be
possible, if salvation lay ready to our hand, and could without great
labor be found, that it should be by almost everybody neglected?
But all excellent things are as difficult as they are rare.